#6800821

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1473 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,5)(3,7)(18,23)(20,21)(24,30)(25,29)(27,28)(31,32);(1,2,4,5)(3,6,7,8)(9,10,12,13)(11,15,16,14)(19,26)(24,31)(27,28)(30,32);(2,5)(6,8)(10,13)(14,15)(17,18,22,25)(19,24,26,31)(20,23,21,29)(27,32,28,30);(1,3,4,7)(2,6,5,8)(17,19)(18,24)(20,28)(21,27)(22,26)(23,30)(25,31)(29,32);(9,11,12,16)(10,14,13,15)(17,20,22,21)(18,23,25,29)(19,27,26,28)(24,32,31,30)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6708017);